Quadratic Equations and Inequalities

2.1 Quadratic Equations and Inequalities
 
Definition of Quadratic Equation
A quadratic equation in general form can be wirtten as
\(ax^2+bx+c=0\)
where \(a\), \(b\) and \(c\) are constants and \(a \neq0\).
 
The image is an educational graphic titled ‘Methods of Solving Quadratic Equations.’ It features two methods: 1. ‘Completing the Square Method’: - Rewrite \( ax^2 + bx + c = 0 \) in the form of \( a(x - h)^2 + k = 0 \). - Solve for \( x \). 2. ‘Formula Method’: - Use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The graphic includes the Pandai logo at the bottom right corner.
 
Roots of Quadratic Equation
Solutions or roots of the quadratic equation \(ax^2+bx+c=0\) are the \(x\)-coordinates of the intersection points between the graph \(y=ax^2+bx+c\) and the \(x\)-axis.
 
Formula for Solving Quadratic Equation

The formula for solving a quadratic equation \(ax^2+bx+c=0\) is given as:

\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

 
Forming Quadratic Equation from Given Roots
Formula for Sum and Product of Roots
  • Formula for sum of roots is given by:

\(\text{Sum of roots}=\alpha+\beta=-\dfrac{b}{a}\)

  • While formula of product of roots is given by:

\(\text{Product of roots}=\alpha\beta=\dfrac{c}{a}\)

Forming Quadratic Equation

The quadratic equation with roots \(\alpha\) and \(\beta\) can be written as:

\(x^2-(\text{sum of roots})x+(\text{product of roots})=0\)

 
Example \(1\)
Question

Solve the equation \(x^2+4x-7=0\) by using completing the square method.

Solution

Based on equation \(x^2+4x-7=0\),

\(a=1\),
\(b=4\),
\(c=-7\).


Move the constant term, \(c\) to the right hand side of the equation,

\(\begin{aligned} x^2+4x-7&=0 \\ x^2+4x&=7. \end{aligned}\)


Add the term \(\left( \dfrac{b}{2} \right)^2\) on the left and right hand side of the equation,

\(\begin{aligned} x^2+4x+\left( \dfrac{4}{2} \right)^2&=7+\left( \dfrac{4}{2} \right)^2 \\ x^2+4x+2^2&=7+2^2\\ (x+2)^2&=11\\ x+2&=\pm \sqrt{11}. \end{aligned}\)

\(x=-5.317\) or \(x=1.317\).

Hence, the solutions of the equation \(x^2+4x-7=0\) are \(-5.317\) and \(1.317\).

 
Example \(2\)
Question

Solve the equation \(2x^2-2x-3=0\) by using formula.

Solution

Based on the equation \(2x^2-2x-3=0\),

\(a=2\),
\(b=-2\),
\(c=-3\).


Use the formula for solving quadratic equation,

\(\begin{aligned} x&=\dfrac{-(-2)\pm \sqrt{(-2)^2-4(2)(-3)}}{2(2)} \\ &=\dfrac{2\pm \sqrt{28}}{4} \end{aligned}\)

\(\begin{aligned} x&=\dfrac{2-\sqrt{28}}{4} \\ &=-0.823 \end{aligned}\) or \(\begin{aligned} x&=\dfrac{2+\sqrt{28}}{4} \\ &=1.823 .\end{aligned}\)

Hence, the solutions of the equation \(2x^2-2x-3=0\) are \(-0.823\) and \(1.823\).

 

Quadratic Equations and Inequalities

2.1 Quadratic Equations and Inequalities
 
Definition of Quadratic Equation
A quadratic equation in general form can be wirtten as
\(ax^2+bx+c=0\)
where \(a\), \(b\) and \(c\) are constants and \(a \neq0\).
 
The image is an educational graphic titled ‘Methods of Solving Quadratic Equations.’ It features two methods: 1. ‘Completing the Square Method’: - Rewrite \( ax^2 + bx + c = 0 \) in the form of \( a(x - h)^2 + k = 0 \). - Solve for \( x \). 2. ‘Formula Method’: - Use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The graphic includes the Pandai logo at the bottom right corner.
 
Roots of Quadratic Equation
Solutions or roots of the quadratic equation \(ax^2+bx+c=0\) are the \(x\)-coordinates of the intersection points between the graph \(y=ax^2+bx+c\) and the \(x\)-axis.
 
Formula for Solving Quadratic Equation

The formula for solving a quadratic equation \(ax^2+bx+c=0\) is given as:

\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

 
Forming Quadratic Equation from Given Roots
Formula for Sum and Product of Roots
  • Formula for sum of roots is given by:

\(\text{Sum of roots}=\alpha+\beta=-\dfrac{b}{a}\)

  • While formula of product of roots is given by:

\(\text{Product of roots}=\alpha\beta=\dfrac{c}{a}\)

Forming Quadratic Equation

The quadratic equation with roots \(\alpha\) and \(\beta\) can be written as:

\(x^2-(\text{sum of roots})x+(\text{product of roots})=0\)

 
Example \(1\)
Question

Solve the equation \(x^2+4x-7=0\) by using completing the square method.

Solution

Based on equation \(x^2+4x-7=0\),

\(a=1\),
\(b=4\),
\(c=-7\).


Move the constant term, \(c\) to the right hand side of the equation,

\(\begin{aligned} x^2+4x-7&=0 \\ x^2+4x&=7. \end{aligned}\)


Add the term \(\left( \dfrac{b}{2} \right)^2\) on the left and right hand side of the equation,

\(\begin{aligned} x^2+4x+\left( \dfrac{4}{2} \right)^2&=7+\left( \dfrac{4}{2} \right)^2 \\ x^2+4x+2^2&=7+2^2\\ (x+2)^2&=11\\ x+2&=\pm \sqrt{11}. \end{aligned}\)

\(x=-5.317\) or \(x=1.317\).

Hence, the solutions of the equation \(x^2+4x-7=0\) are \(-5.317\) and \(1.317\).

 
Example \(2\)
Question

Solve the equation \(2x^2-2x-3=0\) by using formula.

Solution

Based on the equation \(2x^2-2x-3=0\),

\(a=2\),
\(b=-2\),
\(c=-3\).


Use the formula for solving quadratic equation,

\(\begin{aligned} x&=\dfrac{-(-2)\pm \sqrt{(-2)^2-4(2)(-3)}}{2(2)} \\ &=\dfrac{2\pm \sqrt{28}}{4} \end{aligned}\)

\(\begin{aligned} x&=\dfrac{2-\sqrt{28}}{4} \\ &=-0.823 \end{aligned}\) or \(\begin{aligned} x&=\dfrac{2+\sqrt{28}}{4} \\ &=1.823 .\end{aligned}\)

Hence, the solutions of the equation \(2x^2-2x-3=0\) are \(-0.823\) and \(1.823\).